Gauss's Law in 3, 2, and 1 Dimension
Gauss's law relates the electric flux through a closed surface to the total charge enclosed by the surface:

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You can use Gauss's law to determine the charge enclosed inside a closed surface on which the electric field is known. However, Gauss's law is most frequently used to determine the electric field from a symmetric charge distribution.

The simplest case in which Gauss's law can be used to determine the electric field is that in which the charge is localized at a point, a line, or a plane. When the charge is localized at a point, so that the electric field radiates in three-dimensional space, the Gaussian surface is a sphere, and computations can be done in spherical coordinates. Now consider extending all elements of the problem (charge, Gaussian surface, boundary conditions) infinitely along some direction, say along the z axis. In this case, the point has been extended to a line, namely, the z axis, and the resulting electric field has cylindrical symmetry. Consequently, the problem reduces to two dimensions, since the field varies only with x and y, or with and in cylindrical coordinates. A one-dimensional problem may be achieved by extending the problem uniformly in two directions. In this case, the point is extended to a plane, and consequently, it has planar symmetry.

Consider a point charge in three-dimensional space. Symmetry requires the electric field to point directly away from the charge in all directions. To find , the magnitude of the field at distance from the charge, the logical Gaussian surface is a sphere centered at the charge. The electric field is normal to this surface, so the dot product of the electric field and an infinitesimal surface element involves . The flux integral is therefore reduced to , where is the magnitude of the electric field on the Gaussian surface, and is the area of the surface.

Part A
Determine the magnitude by applying Gauss's law.
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Express in terms of some or all of the variables/constants , , and .
 =

Now consider the case that the charge has been extended along the z axis. This is generally called a line charge. The usual variable for a line charge density (charge per unit length) is , and it has units (in the SI system) of coulombs per meter.
Part B
By symmetry, the electric field must point radially outward from the wire at each point; that is, the field lines lie in planes perpendicular to the wire. In solving for the magnitude of the radial electric field produced by a line charge with charge density , one should use a cylindrical Gaussian surface whose axis is the line charge. The length of the cylindrical surface should cancel out of the expression for . Apply Gauss's law to this situation to find an expression for .
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Express in terms of some or all of the variables , , and any needed constants.
 =

Now consider the case with one effective direction. In order to make a problem effectively one-dimensional, it is necessary to extend a charge to infinity along two orthogonal axes, conventionally taken to be x and y. When the charge is extended to infinity in the xy plane (so that by symmetry, the electric field will be directed in the z direction and depend only on z), the charge distribution is sometimes called a sheet charge. The symbol usually used for two-dimensional charge density is either , or . In this problem we will use . has units of coulombs per meter squared.
Part C
In solving for the magnitude of the electric field produced by a sheet charge with charge density , one may use a Gaussian surface in the shape of a rectangular box two of whose faces lie a distance above and below the sheet of charge. The area of these faces must then be calculated; they will cancel out of the expression for in the end. The result of applying Gauss's law to this situation then gives an expression for for both and .
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Express for in terms of some or all of the variables/constants , , and .